**- A New Paradigm -**

"God does not play dice"

*vs*"God does not waste computational resources"

Drunken walks are ideal for map generation for 3 reasons:

- The generated map is inherently fractal.
- The generated map is definitively random.
- The map can be generated dynamically (on the fly). This dynamic map generation allows for unlimited map size (boundless and infinite).

Drunken walks can be biased to favor any combination of the 6 cardinal/absolute directions (NW, NE, East, SE, SW, West) and the 6 relative directions (straight forward = 12:00, right forward = 2:00, right backward = 4:00, straight backward = 6:00, left backward = 8:00, left forward = 10:00).

These biases can be encoded in 6 weights for cardinal/absolute directions, and 6 weights for relative directions (relative to the current direction of movement).

A drunken walk pattern can thus be defined by 12 weights, and we call this collection of weights the "Direction Tensor". The word "tensor" is used here to mean roughly "an object whose shadow is a vector". At any given point in time, the likelihood of moving in each direction is proportional to the product of the cardinal/absolute weight, and the corresponding relative weight (each relative direction corresponds to an absolute direction, with respect to the current direction of movement).

The effects of chaging the 6 cardinal/absolute weights are fairly predictable, so this study examines only the effects of relative weightings (the 6 cardinal/absolute weights are taken to be equal to each other, so there are no cardinal biases).

**- Example Patterns -**

Limiting movement to backwards right and left (4:00 and 8:00) leads to a relatively compact but elongated form:

50% right backwards (4:00), 50% left backwards (8:00). |

66% right backwards (4:00), 33% left backwards (8:00). |

66% straight forward (12:00), 33% right forward (2:00). |

50% right forward (2:00), 50% left forward (10:00). [example 1/2] |

50% right forward (2:00), 50% left forward (10:00). [example 2/2] |

20% straight forward (12:00), 20% right forward (2:00), 20% right backward (4:00), 20% left backward (8:00), 20% left forward (10:00). [example 1/3] |

20% straight forward (12:00), 20% right forward (2:00), 20% right backward (4:00), 20% left backward (8:00), 20% left forward (10:00). [example 2/3] |

20% straight forward (12:00), 20% right forward (2:00), 20% right backward (4:00), 20% left backward (8:00), 20% left forward (10:00). [example 3/3] |

**- VBA Code (Class Module) -**

'6 weights for the 6 absolute directions:

Public NW As Single

Public NE As Single

Public East As Single

Public SE As Single

Public SW As Single

Public West As Single

'6 weights for the 6 relative directions:

Public StraightForward As Single

Public RightForward As Single

Public RightBackward As Single

Public StraightBackward As Single

Public LeftBackward As Single

Public LeftForward As Single

'6 numbers for current fractional likelihood of each absolute direction:

Public ProbabilityNW As Single

Public ProbabilityNE As Single

Public ProbabilityEast As Single

Public ProbabilitySE As Single

Public ProbabilitySW As Single

Public ProbabilityWest As Single

Public Function GetDirection(ByVal Previous As Location, ByVal Current As Location)

Dim Direction As String

Dim SouthShift, EastShift As Integer

SouthShift = Current.Row - Previous.Row

EastShift = Current.Col - Previous.Col

Direction = "resting"

If SouthShift = 0 And EastShift = 1 Then Direction = "East"

If SouthShift = 0 And EastShift = -1 Then Direction = "West"

If SouthShift = 1 And EastShift = 0 Then Direction = "SE"

If SouthShift = 1 And EastShift = -1 Then Direction = "SW"

If SouthShift = -1 And EastShift = 1 Then Direction = "NE"

If SouthShift = -1 And EastShift = 0 Then Direction = "NW"

GetDirection = Direction

End Function

Public Sub CalculateProbabilities(ByVal Direction As String)

'The public variable "Direction" should be set before calling this sub.

Dim AbsoluteWeights(6) As Single

Dim RelativeWeights(6) As Single

Dim CombinedWeights(6) As Single

Dim Counter, RotatingCounter As Integer

Dim TotalWeights As Single

AbsoluteWeights(0) = Me.NW

AbsoluteWeights(1) = Me.NE

AbsoluteWeights(2) = Me.East

AbsoluteWeights(3) = Me.SE

AbsoluteWeights(4) = Me.SW

AbsoluteWeights(5) = Me.West

RelativeWeights(0) = Me.StraightForward

RelativeWeights(1) = Me.RightForward

RelativeWeights(2) = Me.RightBackward

RelativeWeights(3) = Me.StraightBackward

RelativeWeights(4) = Me.LeftBackward

RelativeWeights(5) = Me.LeftForward

TotalWeights = 0

'Rotate RelativeWeights depending on Direction:

Select Case Direction

Case "NW"

For Counter = 0 To 5

CombinedWeights(Counter) = AbsoluteWeights(Counter) * RelativeWeights(Counter)

TotalWeights = TotalWeights + CombinedWeights(Counter)

Next Counter

Case "NE"

For Counter = 0 To 5

RotatingCounter = (Counter - 1) Mod 6

'We need the next line, because vba does "mod" wrong.

If RotatingCounter < 0 Then RotatingCounter = RotatingCounter + 6

CombinedWeights(Counter) = AbsoluteWeights(Counter) * RelativeWeights(RotatingCounter)

TotalWeights = TotalWeights + CombinedWeights(Counter)

Next Counter

Case "East"

For Counter = 0 To 5

RotatingCounter = (Counter - 2) Mod 6

If RotatingCounter < 0 Then RotatingCounter = RotatingCounter + 6

CombinedWeights(Counter) = AbsoluteWeights(Counter) * RelativeWeights(RotatingCounter)

TotalWeights = TotalWeights + CombinedWeights(Counter)

Next Counter

Case "SE"

For Counter = 0 To 5

RotatingCounter = (Counter - 3) Mod 6

If RotatingCounter < 0 Then RotatingCounter = RotatingCounter + 6

CombinedWeights(Counter) = AbsoluteWeights(Counter) * RelativeWeights(RotatingCounter)

TotalWeights = TotalWeights + CombinedWeights(Counter)

Next Counter

Case "SW"

For Counter = 0 To 5

RotatingCounter = (Counter - 4) Mod 6

If RotatingCounter < 0 Then RotatingCounter = RotatingCounter + 6

CombinedWeights(Counter) = AbsoluteWeights(Counter) * RelativeWeights(RotatingCounter)

TotalWeights = TotalWeights + CombinedWeights(Counter)

Next Counter

Case "West"

For Counter = 0 To 5

RotatingCounter = (Counter - 5) Mod 6

If RotatingCounter < 0 Then RotatingCounter = RotatingCounter + 6

CombinedWeights(Counter) = AbsoluteWeights(Counter) * RelativeWeights(RotatingCounter)

TotalWeights = TotalWeights + CombinedWeights(Counter)

Next Counter

Case "resting"

'If the animal rested on the last turn, absolute/cardinal movement bias controls:

For Counter = 0 To 5

CombinedWeights(Counter) = AbsoluteWeights(Counter)

TotalWeights = TotalWeights + CombinedWeights(Counter)

Next Counter

End Select

'Normalize the probabilities so they add up to 1:

If TotalWeights <> 0 Then

For Counter = 0 To 5

CombinedWeights(Counter) = CombinedWeights(Counter) / TotalWeights

Next Counter

End If

Me.ProbabilityNW = CombinedWeights(0)

Me.ProbabilityNE = CombinedWeights(1)

Me.ProbabilityEast = CombinedWeights(2)

Me.ProbabilitySE = CombinedWeights(3)

Me.ProbabilitySW = CombinedWeights(4)

Me.ProbabilityWest = CombinedWeights(5)

End Sub

Public Sub NormalizeProbabilities(ByVal TotalWeights As Single)

'Use zero to indicate "recalculate TotalWeights":

If TotalWeights = 0 Then

TotalWeights = Me.ProbabilityNW + Me.ProbabilityNE + Me.ProbabilityEast _

+ Me.ProbabilitySE + Me.ProbabilitySW + Me.ProbabilityWest

End If

'If it actually is zero (based on math), then use "absolute values":

If TotalWeights = 0 Then

Me.CalculateProbabilities ("resting")

Else

Me.ProbabilityNW = Me.ProbabilityNW / TotalWeights

Me.ProbabilityNE = Me.ProbabilityNE / TotalWeights

Me.ProbabilityEast = Me.ProbabilityEast / TotalWeights

Me.ProbabilitySE = Me.ProbabilitySE / TotalWeights

Me.ProbabilitySW = Me.ProbabilitySW / TotalWeights

Me.ProbabilityWest = Me.ProbabilityWest / TotalWeights

End If

End Sub

Copyright 2016 Matthew J Curran. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this code except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.